Heat milk until warm but not hot. In a large bowl combine with yeast, mix a little then let sit for 5 minutes until mixture is foamy.
Beat eggs, butter, sugar, and salt into the yeast mixture. Add half the flour and mix until combined. Then mix in the rest of the flour until dough pulls away from sides of bowl. Knead for a few minutes on a floured surface. Cover the inside of a large bowl with oil. Put dough in large bowl and cover with dish towel. Let rise for about and hour until it is double in size.
Turn dough out onto a well floured surface and roll out into 1/2 inch thickness. Use cookie cutters or glasses to cut doughnuts. Shot glasses work great as the center cut out. Place cut doughnuts on floured baking sheets with plenty of space in between. Cover with kitchen towels and let sit at room temperature for about 45 minutes or until doubled in size.
About 15 minutes before doughnuts are done rising, place pot with oil over medium heat. Get the oil to 375 degrees use a meat thermometer to make sure. Line cooling racks with paper towels.
Use a slotted spoon to put a few doughnuts in the pot. Let cook for around 45 seconds and then flip and cook until golden. Take out and let rest on paper towels before icing.
Prepare icing by mixing powdered sugar, maple syrup, milk and vanilla.
Dip in icing and let rest on wire racks before serving.